Data structure: \(O = (W, A, Y)\)
Underlying data generating process, \(P_{U,X}\)
Outcome of interest: \(E_0[Y|a,W]\), \(a \in [0,5]\), the causal dose-response curve
## W A Y
## Min. :-3.851249 Min. :0.0000 Min. :0.000
## 1st Qu.:-0.660181 1st Qu.:0.6267 1st Qu.:0.000
## Median :-0.004693 Median :2.0063 Median :0.000
## Mean : 0.001261 Mean :2.1367 Mean :0.467
## 3rd Qu.: 0.678899 3rd Qu.:3.4288 3rd Qu.:1.000
## Max. : 3.492751 Max. :5.0000 Max. :1.000
## Summary of A given W < -1:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.000 1.422 2.769 2.720 4.168 5.000
## Summary of A given -1 < W <= 0:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.0000 0.9032 2.2570 2.3275 3.6314 5.0000
## Summary of A given 0 < W <= 1:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.0000 0.4132 1.7328 1.9460 3.1717 5.0000
## Summary of A given 1 < W:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.000 0.000 1.246 1.552 2.587 5.000
Data structure: \(O = (W, A, Y)\)
Underlying data generating process, \(P_{U,X}\)
Outcome of interest: \(E_0[Y|a,W]\), \(a \in [0,5]\), the causal dose-response curve
## W A Y
## Min. :-3.967532 Min. :0.000 Min. :0.0000
## 1st Qu.:-0.669179 1st Qu.:0.616 1st Qu.:0.0000
## Median :-0.006594 Median :1.989 Median :0.0000
## Mean :-0.002222 Mean :2.121 Mean :0.0693
## 3rd Qu.: 0.662488 3rd Qu.:3.407 3rd Qu.:0.0000
## Max. : 3.604661 Max. :5.000 Max. :1.0000
## Summary of A given W < -1:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.000 1.314 2.821 2.713 4.229 5.000
## Summary of A given -1 < W <= 0:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.0000 0.9032 2.2378 2.2838 3.5098 5.0000
## Summary of A given 0 < W <= 1:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.0000 0.4145 1.7833 1.9454 3.1190 5.0000
## Summary of A given 1 < W:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.000 0.000 1.237 1.561 2.598 5.000
## The average lambda of CV-HAL: 0.0036 (= 1 * lambda_CV )
## The average lambda of globally undersmoothed HAL: 0.0016 (= 0.3371 * lambda_CV )
## The average lambdatime of locally undersmoothed HAL: 0.0036, 0.0035, 0.0034, 0.0034, 0.0034, 0.0034, 0.0034, 0.0034, 0.0035, 0.0034, 0.0034 (= [1, 0.9924, 0.9842, 0.9853, 0.9863, 0.9869, 0.9887, 0.9887, 0.9897, 0.9824, 0.9841] * lambda_CV
## The average fitting time for CV-HAL: 4.9046 seconds
## The average fitting time for globally undersmoothed HAL: 6.5215 seconds
## The average fitting time for locally undersmoothed HAL: 19.8505 seconds
Data structure: \(O = (W, A, Y)\)
Underlying data generating process, \(P_{U,X}\)
Outcome of interest: \(E_0[Y|a,W]\), \(a \in [0,5]\), the causal dose-response curve
## W A Y
## Min. :-3.999772 Min. :0.000 Min. :0.0000
## 1st Qu.:-0.677979 1st Qu.:0.611 1st Qu.:0.0000
## Median :-0.009506 Median :2.017 Median :0.0000
## Mean : 0.006333 Mean :2.125 Mean :0.4319
## 3rd Qu.: 0.693805 3rd Qu.:3.398 3rd Qu.:1.0000
## Max. : 4.136862 Max. :5.000 Max. :1.0000
## Summary of A given W < -1:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.000 1.394 2.834 2.716 4.151 5.000
## Summary of A given -1 < W <= 0:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.000 0.882 2.208 2.284 3.614 5.000
## Summary of A given 0 < W <= 1:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.0000 0.3999 1.7882 1.9479 3.1441 5.0000
## Summary of A given 1 < W:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.000 0.000 1.318 1.574 2.612 5.000
## The average lambda of CV-HAL: 0.0026 (= 1 * lambda_CV )
## The average lambda of globally undersmoothed HAL: 0.0009 (= 0.2837 * lambda_CV )
## The average lambdatime of locally undersmoothed HAL: 0.0024, 0.0024, 0.0024, 0.0024, 0.0025, 0.0025, 0.0025, 0.0024, 0.0025, 0.0025, 0.0026 (= [0.8706, 0.8682, 0.8726, 0.8927, 0.9219, 0.9584, 0.9378, 0.9093, 0.9588, 0.9761, 0.9854] * lambda_CV
## The average fitting time for CV-HAL: 3.7293 seconds
## The average fitting time for globally undersmoothed HAL: 6.0426 seconds
## The average fitting time for locally undersmoothed HAL: 24.0037 seconds
Data structure: \(O = (W_1, W_2, W_3, W_4, W_5, A, Y)\)
Underlying data generating process, \(P_{U,X}\)
Outcome of interest: \(E_0[Y|a,W]\), \(a \in [0,5]\), the causal dose-response curve
## W1 W2 W3 W4
## Min. :-4.254523 Min. :-4.697934 Min. :-1.584 Min. :-1.2658
## 1st Qu.:-0.668564 1st Qu.:-0.679257 1st Qu.: 3.987 1st Qu.: 0.6482
## Median :-0.009893 Median :-0.010669 Median : 4.983 Median : 0.9946
## Mean :-0.006976 Mean :-0.004105 Mean : 4.986 Mean : 0.9964
## 3rd Qu.: 0.654261 3rd Qu.: 0.670930 3rd Qu.: 5.987 3rd Qu.: 1.3502
## Max. : 4.166622 Max. : 4.143946 Max. :10.572 Max. : 2.9153
## W5 A Y
## Min. :-0.3376 Min. :0.000 Min. :0.0000
## 1st Qu.: 0.7592 1st Qu.:1.592 1st Qu.:0.0000
## Median : 0.9945 Median :2.534 Median :1.0000
## Mean : 0.9955 Mean :2.543 Mean :0.6249
## 3rd Qu.: 1.2275 3rd Qu.:3.508 3rd Qu.:1.0000
## Max. : 2.2710 Max. :5.000 Max. :1.0000
Data structure: \(O = (W_1, W_2, W_3, W_4, W_5, A, Y)\)
Underlying data generating process, \(P_{U,X}\)
Outcome of interest: \(E_0[Y|a,W]\), \(a \in [0,5]\), the causal dose-response curve
## W1 W2 W3 W4
## Min. :-4.10305 Min. :-3.992762 Min. :-0.723 Min. :-1.0645
## 1st Qu.:-0.66856 1st Qu.:-0.665766 1st Qu.: 3.993 1st Qu.: 0.6574
## Median : 0.01409 Median : 0.004221 Median : 5.007 Median : 0.9974
## Mean : 0.01345 Mean : 0.004542 Mean : 5.008 Mean : 0.9924
## 3rd Qu.: 0.68434 3rd Qu.: 0.676313 3rd Qu.: 6.014 3rd Qu.: 1.3343
## Max. : 3.37515 Max. : 4.074948 Max. :10.222 Max. : 2.8006
## W5 A Y
## Min. :-0.1632 Min. :0.000 Min. :0.000
## 1st Qu.: 0.7565 1st Qu.:1.604 1st Qu.:0.000
## Median : 0.9909 Median :2.561 Median :0.000
## Mean : 0.9929 Mean :2.554 Mean :0.476
## 3rd Qu.: 1.2283 3rd Qu.:3.509 3rd Qu.:1.000
## Max. : 2.2176 Max. :5.000 Max. :1.000
Data structure: \(O = (W_1, W_2, W_3, A, Y)\)
Underlying data generating process, \(P_{U,X}\)
Outcome of interest: \(E_0[Y|a,W]\), \(a \in (0,1])\), the causal dose-response curve
## W1 W2 W3 A
## Min. :0.0000212 Min. :0.0000 Min. :-3.20646 Min. :0.008172
## 1st Qu.:0.2550912 1st Qu.:0.0000 1st Qu.: 0.07964 1st Qu.:0.483313
## Median :0.5016742 Median :1.0000 Median : 0.80953 Median :0.661615
## Mean :0.5026247 Mean :0.6984 Mean : 0.80890 Mean :0.632873
## 3rd Qu.:0.7530768 3rd Qu.:1.0000 3rd Qu.: 1.53093 3rd Qu.:0.807451
## Max. :0.9999523 Max. :1.0000 Max. : 5.10017 Max. :0.999333
## Y
## Min. :0.000
## 1st Qu.:0.000
## Median :0.000
## Mean :0.483
## 3rd Qu.:1.000
## Max. :1.000
## The average lambda of CV-HAL: 0.0090 (= 1 * lambda_CV )
## The average lambda of globally undersmoothed HAL: 0.0023 (= 0.2632 * lambda_CV )
## The average lambdatime of locally undersmoothed HAL: 0.006, 0.0058, 0.0057, 0.0057, 0.0056, 0.0056, 0.0056, 0.0054, 0.0053, 0.0051, 0.0049, 0.0044, 0.0043, 0.0043, 0.0042, 0.0045, 0.0045 (= [0.6913, 0.676, 0.6703, 0.666, 0.6611, 0.6609, 0.6546, 0.6404, 0.6239, 0.603, 0.5776, 0.5095, 0.4942, 0.4902, 0.4755, 0.5031, 0.5107] * lambda_CV
## The average fitting time for CV-HAL: 40.4222 seconds
## The average fitting time for globally undersmoothed HAL: 46.4000 seconds
## The average fitting time for locally undersmoothed HAL: 159.3079 seconds